Stability of Vertical Rotations of an Axisymmetric Ellipsoid on a Vibrating Plane
In this paper, we address the problem of an ellipsoid with axisymmetric mass distribution rolling on a horizontal absolutely rough plane under the assumption that the supporting plane performs periodic vertical oscillations. In the general case, the problem reduces to a system with one and a half de...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/18/3948 |
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| author | Alexander A. Kilin Elena N. Pivovarova |
| author_facet | Alexander A. Kilin Elena N. Pivovarova |
| author_sort | Alexander A. Kilin |
| collection | DOAJ |
| description | In this paper, we address the problem of an ellipsoid with axisymmetric mass distribution rolling on a horizontal absolutely rough plane under the assumption that the supporting plane performs periodic vertical oscillations. In the general case, the problem reduces to a system with one and a half degrees of freedom. In this paper, instead of considering exact equations, we use a vibrational potential that describes approximately the dynamics of a rigid body on a vibrating plane. Since the vibrational potential is invariant under rotation about the vertical, the resulting problem with the additional potential is integrable. For this problem, we analyze the influence of vibrations on the linear stability of vertical rotations of the ellipsoid. |
| first_indexed | 2024-03-10T22:30:17Z |
| format | Article |
| id | doaj.art-ac745b96c1be468f84e0370feb8c57e0 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T22:30:17Z |
| publishDate | 2023-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-ac745b96c1be468f84e0370feb8c57e02023-11-19T11:49:48ZengMDPI AGMathematics2227-73902023-09-011118394810.3390/math11183948Stability of Vertical Rotations of an Axisymmetric Ellipsoid on a Vibrating PlaneAlexander A. Kilin0Elena N. Pivovarova1Ural Mathematical Center, Udmurt State University, Izhevsk 426034, RussiaUral Mathematical Center, Udmurt State University, Izhevsk 426034, RussiaIn this paper, we address the problem of an ellipsoid with axisymmetric mass distribution rolling on a horizontal absolutely rough plane under the assumption that the supporting plane performs periodic vertical oscillations. In the general case, the problem reduces to a system with one and a half degrees of freedom. In this paper, instead of considering exact equations, we use a vibrational potential that describes approximately the dynamics of a rigid body on a vibrating plane. Since the vibrational potential is invariant under rotation about the vertical, the resulting problem with the additional potential is integrable. For this problem, we analyze the influence of vibrations on the linear stability of vertical rotations of the ellipsoid.https://www.mdpi.com/2227-7390/11/18/3948axisymmetric ellipsoidvibrating planenonholonomic constraintpermanent rotationsvertical rotationsstability |
| spellingShingle | Alexander A. Kilin Elena N. Pivovarova Stability of Vertical Rotations of an Axisymmetric Ellipsoid on a Vibrating Plane Mathematics axisymmetric ellipsoid vibrating plane nonholonomic constraint permanent rotations vertical rotations stability |
| title | Stability of Vertical Rotations of an Axisymmetric Ellipsoid on a Vibrating Plane |
| title_full | Stability of Vertical Rotations of an Axisymmetric Ellipsoid on a Vibrating Plane |
| title_fullStr | Stability of Vertical Rotations of an Axisymmetric Ellipsoid on a Vibrating Plane |
| title_full_unstemmed | Stability of Vertical Rotations of an Axisymmetric Ellipsoid on a Vibrating Plane |
| title_short | Stability of Vertical Rotations of an Axisymmetric Ellipsoid on a Vibrating Plane |
| title_sort | stability of vertical rotations of an axisymmetric ellipsoid on a vibrating plane |
| topic | axisymmetric ellipsoid vibrating plane nonholonomic constraint permanent rotations vertical rotations stability |
| url | https://www.mdpi.com/2227-7390/11/18/3948 |
| work_keys_str_mv | AT alexanderakilin stabilityofverticalrotationsofanaxisymmetricellipsoidonavibratingplane AT elenanpivovarova stabilityofverticalrotationsofanaxisymmetricellipsoidonavibratingplane |